**2. Mathematical Model**

The unsteady two-dimensional stagnation-point flow of a hybrid Al2O3-Cu/H2O nanofluid over a convectively heated stretching/shrinking sheet with the influence of velocity slip is considered in this research work, as illustrated in Figure 1 (see Dzulkifli et al. [59]). The stretching/shrinking velocity is denoted by *uw*(*<sup>x</sup>*, *t*) = *bx*/(1 − *ct*), where *b* denotes a constant corresponds to stretching (*b* > 0) and shrinking (*b* < 0) cases while *c* signifies the unsteadiness problem and *ue*(*<sup>x</sup>*, *t*) = *ax*/(<sup>1</sup> − *ct*) is the velocity of the free stream where *a* > 0 represents the strength of the stagnation flow. The ambient temperature and the reference temperature are *T*∞ and *T*0, respectively. Now, we let the bottom of the sheet be heated by convection from a hot fluid at a specific temperature *Tf*(*<sup>x</sup>*, *t*) = *T*∞ +*T*0 *ax*<sup>2</sup> 2ν (1 − *ct*)−3/2 which supplies a coefficient of heat transfer, expressed by *hf* . From all of the assumptions above; the governing boundary layer equations can be acknowledged as [34].

$$
\frac{\partial \mu}{\partial x} + \frac{\partial v}{\partial y} = 0,
\tag{1}
$$

$$
\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \frac{\partial \mathbf{u}}{\partial \mathbf{x}} + \mathbf{v} \frac{\partial \mathbf{u}}{\partial y} = \frac{\partial \mathbf{u}\_{\text{e}}}{\partial t} + \mathbf{u}\_{\text{e}} \frac{\partial \mathbf{u}\_{\text{e}}}{\partial \mathbf{x}} + \frac{\mu\_{\text{huf}}}{\rho\_{\text{huf}}} \frac{\partial^2 \mathbf{u}}{\partial y^2} \tag{2}
$$

$$
\rho \frac{\partial T}{\partial t} + \mu \frac{\partial T}{\partial \mathbf{x}} + v \frac{\partial T}{\partial y} = \frac{k\_{\text{luuf}}}{\left(\rho \mathbb{C}\_p\right)\_{\text{luuf}}} \frac{\partial^2 T}{\partial y^2} \,\tag{3}
$$

where *u* denotes the component of velocity in *x*− axis, *v* is the velocity component in *y*− axis, μ*hn f* is the Al2O3-Cu/H2O dynamic viscosity, ρ*hn f* the density of Al2O3-Cu/H2O, *T* is the Al2O3-Cu/H2O temperature, *khn f* is the thermal/heat conductivity of Al2O3-Cu/H2O and <sup>ρ</sup>*Cphn f* is the Al2O3-Cu/H2O heat capacity. The boundary conditions, together with the partial slip for velocity, are set to

$$\begin{aligned} \mu = \mathfrak{u}\_{\mathfrak{w}}(\mathbf{x}, t) + H\_1 \nu \frac{\partial \mathfrak{u}}{\partial y'} \quad & \mathbf{v} = \mathbf{0}, \quad -k\_{\mathrm{Im}f} \frac{\partial T}{\partial y} = h\_f + \begin{pmatrix} T\_f - T \end{pmatrix} \quad \text{at} \quad \mathcal{Y} = \mathbf{0}, \\\ \mathbf{u} \rightarrow \mathfrak{u}\_{\mathfrak{t}}(\mathbf{x}, t), \quad T \rightarrow T\_{\mathrm{co}} \quad \text{as} \quad \mathcal{Y} \rightarrow \infty, \end{aligned} \tag{4}$$

where *H*1 = *H*(1 − *ct*)1/2 is the velocity slip factor, in which *H* refers to the initial value of the velocity slip factor. The copper (Cu) thermophysical properties, along with aluminum oxide (Al2O3) and water (H2O) nanoparticles, are provided in Table 1, as demonstrated by [60]. In the meantime, Table 2 issued the thermophysical properties hybrid nanofluid as established by [53,57]. The nanoparticles solid volume fraction is represented by φ, ρ*f* indicates the H2O density, and ρ*s* is the density of the hybrid nanoparticle, *Cp* is the constant pressure of heat capacity, while *k f* denotes the thermal conductivity of H2O and *ks* is the hybrid nanoparticles thermal conductivity.

**Figure 1.** The schematic of problem flow (Dzulkifli et al. [59]).


**Table 1.** Cu thermophysical properties along with Al2O3 and H2O (Oztop and Abu Nada [60]).

**Table 2.** Hybrid Al2O3-Cu/H2O nanofluids thermophysical properties (Takabi and Salehi [53], Ghalambaz et al. [57]).


In order to express the governing Equations (1)–(3) concerning the boundary conditions (4) in a much simpler form, the subsequent similarity transformations are presented [34]

$$\psi = \left(\frac{a\nu}{1 - ct}\right)^{1/2} \text{xf}(\eta), \theta(\eta) = \frac{T - T\_{\infty}}{T\_f - T\_{\infty}}, \eta = \left(\frac{a}{\nu(1 - ct)}\right)^{1/2} y,\tag{5}$$

where ψ is the stream function that can be specified as *u* = ∂ψ/∂*y*, *v* = −∂ψ/∂*y* and η is the similarity variable. Thus, we attain

$$u = \frac{a\mathbf{x}}{(1 - ct)} f'(\eta), \upsilon = -\left(\frac{a\nu}{1 - ct}\right)^{1/2} f(\eta). \tag{6}$$

In view of the above relations, by employing the similarity variables (5) and (6), Equations (2) and (3) reduce to the following set of nonlinear similarity differential equations

$$\frac{\mu\_{\text{Im}f}/\mu\_f}{\rho\_{\text{Im}f}/\rho\_f} f'''' + f f'' - f'^2 + 1 - \varepsilon \left( f' + \frac{1}{2} \eta f'' - 1 \right) = 0,\tag{7}$$

$$\frac{1}{\Pr} \frac{k\_{\rm Imf} / k\_f}{\left(\rho \mathbb{C}\_p\right)\_{\rm Imf} / \left(\rho \mathbb{C}\_p\right)\_f} \theta'' + f \theta' - 2f' \theta + \frac{\varepsilon}{2} (\eta \theta' + 3\theta) = 0. \tag{8}$$

Here, ε measures the unsteadiness parameter with ε = *<sup>c</sup>*/*<sup>a</sup>*, Pr represents the Prandtl number where Pr = <sup>ν</sup>*f* /<sup>α</sup>*f* . Next, the initial and boundary conditions (4) now transform into

$$\begin{aligned} f(0) = 0, f'(0) = \lambda + \gamma f''(0), \, -\frac{k\_{\rm inf}}{k\_f} \theta'(0) = \text{Bi}[1 - \theta(0)], \\\ f'(\eta) \to 1, \theta(\eta) \to 0, \text{ while } \eta \to \infty. \end{aligned} \tag{9}$$

From Equation (9), λ symbolises as the ratio of velocity parameter, γ and Bi are the dimensionless velocity slip parameter and Biot number, respectively, which are described as

$$
\lambda = \frac{b}{a'} \gamma = H(a\nu)^{1/2}, \text{Bi} = \frac{\text{h}\_f}{k\_f} \sqrt{\frac{\nu(1-ct)}{a}}.\tag{10}
$$

Next, we define the skin friction coefficient -*Cf*and the local Nusselt number (*Nux*) as

$$\mathcal{C}\_f = \frac{\tau\_w}{\rho\_f \mathcal{U}\_\mathfrak{e}^{2}}, \; \mathcal{N}\mathfrak{u}\_\mathfrak{x} = \frac{\mathbf{x}q\_w}{k\_f(T\_f - T\_\infty)}.\tag{11}$$

The shear stress along the *x*− direction is represented by τ*w*, while *qw* signifies the surface heat flux that accentuated by

$$
\pi\_{w} = \mu\_{\text{luuf}} \left( \frac{\partial u}{\partial y} \right)\_{y=0} \quad q\_{w} = -k\_{\text{luuf}} \left( \frac{\partial T}{\partial y} \right)\_{y=0} . \tag{12}
$$

By applying (5) and (12) into (11), we acquire

$$
\sqrt{\text{Re}\_x}\mathbb{C}\_f = \frac{\mu\_{\text{lnf}}}{\mu\_f} f''(0), \quad \frac{1}{\sqrt{\text{Re}\_x}}\text{Nu}\_x = -\frac{k\_{\text{lnf}}}{k\_f} \theta'(0), \tag{13}
$$

provided that Re*x* = *uex* <sup>ν</sup>*f* is the local Reynolds number in *x*- axis.
